Fig. 1. Strongly connected dependency graph G f =(V f , E f , π f ) with loop number L G f (V f )=6 of a 24-dimensional Boolean monomial dynamical system f ∈ MF 24 24 (F 2 ). Circles (blue) demarcate each of the six loop equivalence classes. Essentially, the dependency graph is a closed path of length 6. and because of (2) in the previous theorem clearly  a =  b∈  a N t (b) Given one loop equivalence class  a ⊆ V G , the set of all the t loop equivalence classes can be ordered in the following manner  a i :=  a,  a i+1 =  b∈  a i N 1 (b),  a i+j =  b∈  a i N j (b),  a i+t−1 =  b∈  a i N t−1 (b) For any c ∈  b∈  a i N t−1 (b) it must hold N 1 (c) ⊆  a i (if N 1 (c) ∩  a j = ∅ with j = i, then  a i =  a j ). Thus, the graph G can be visualized as (see Fig. 1)  a i ⇒  a i+1 ⇒ ···⇒  a i+j ⇒  a (i+j+1) mod t ⇒ ⇒  a i+t−1 ⇒  a (i+t ) mod t Due to the fact  a =  b∈  a N t (b) ∀ a ∈ V G , we can conclude that the claims of the previous lemma still hold if the sequence lengths m and m  are replaced by the more general lengths λt + m and λ  t + m  , where λ, λ  ∈ N. 3.2 Boolean monomial control systems: Control theoretic questions studied We start this section with the formal definition of a time invariant monomial control system over a finite field. Using the results stated in the previous section, we provide a very compact nomenclature for such systems. After further elucidations, and, in particular, after providing the formal definition of a monomial feedback controller, we clearly state the main control theoretic problem to be studied in Section 3.3 of this chapter. Definition 54. Let F q be a finite field, n ∈ N a natural number and m ∈ N 0 a nonnegative integer. A mapping g : F n q × F m q → F n q is called time invariant monomial control system over F q if for 469 Discrete Time Systems with Event-Based Dynamics: Recent Developments in Analysis and Synthesis Methods every i ∈{1, , n} there are two tuples (A i1 , , A in ) ∈ E n q and (B i1 , , B im ) ∈ E m q such that g i (x, u)=x A i1 1 x A in n u B i1 1 u B im m ∀ (x, u) ∈ F n q ×F m q Remark 55. In the case m = 0, we have F m q = F 0 q = {()} (the set containing the empty tuple) and thus F n q × F m q = F n q × F 0 q = F n q ×{()} = F n q . In other words, g is a monomial dynamical system over F q . From now on we will refer to a time invariant monomial control system over F q as monomial control system over F q . Definition 56. Let X be a nonempty finite set and n, l ∈ N natural numbers. The set of all functions f : X l → X n is denoted with F n l (X). Definition 57. Let F q be a finite field and l, m, n ∈ N natural numbers. Furthermore, let E q be the exponents semiring of F q and M(n ×l; E q ) the set of n × l matrices with entries in E q . Consider the map Γ : F l m (F q ) × M(n ×l; E q ) → F n m (F q ) ( f , A) → Γ A ( f) where Γ A ( f) is defined for every x ∈ F m q and i ∈{1, , n} by Γ A ( f)(x) i := f 1 (x) A i1 f l (x) A il We denote the mapping Γ A ( f ) ∈ F n m (F q ) simply A f . Remark 58. Let l = m, id ∈ F m m (F q ) be the identity map (i.e. id i (x)=x i ∀ i ∈{1, , m}) and A ∈ M(n × m; E q ) Then the following relationship between the mapping Aid ∈ F n m (F q ) and any f ∈ F m m (F q ) holds Aid ( f (x)) = Af(x) ∀ x ∈ F m q Remark 59. Consider the case l = m = n. For every monomial dynamical system f ∈ MF n n (F q ) ⊂ F n n (F q ) with corresponding matrix F := Ψ −1 ( f ) ∈ M(n × n; E q ) it holds Fid = f . On the other hand, given a matrix F ∈ M(n × n; E q ) we have Ψ −1 (Fid)=F. Moreover, the map Γ : F n n (F q ) × M(n ×n; E q ) → F n n (F q ) is an action of the multiplicative monoid M(n ×n; E q ) on the set F n n (F q ). It holds namely, that 12 If = f ∀ f ∈ F n n (F q ) (which is trivial) and (A · B) f = A(Bf) ∀ f ∈ F n n (F q ), A, B ∈ M(n ×n; E q ). To see this, consider (( A · B) f ) i (x)= f 1 (x) (A·B) i1 f n (x) (A·B) in = n ∏ j=1 f j (x) (A i1 •B 1j ⊕ ⊕A in •B nj ) =(Aid ◦ Bid ) i ( f (x)) = ( Ai d) i (Bid( f (x))) =( Aid) i ( fB(x)) = ( A(Bf)) i (x) where id ∈ F n n (F q ) is the identity map (i.e. id i (x)=x i ∀ i ∈{1, , n}). (cf. with the proof of Theorem 29). As a consequence, MF n n (F q ) is the orbit in F n n (F q ) of id under the monoid M(n × n; E q ). In particular (see Theorem 29), we have (F · G)id = F(Gid)= f ◦ g 12 I ∈ M(n ×n; E q ) denotes the identity matrix. 470 Discrete Time Systems where g ∈ MF n n (F q ) is another monomial dynamical system with corresponding matrix G := Ψ −1 (g) ∈ M(n × n; E q ). Lemma 60. Let F q be a finite field, n ∈ N a natural number and m ∈ N 0 a nonnegative integer. Furthermore, let id ∈ F (n+m ) ( n+m) (F q ) be the identity map (i.e. id i (x)=x i ∀ i ∈{1, , n + m}) and g : F n q ×F m q → F n q a monomial control system over F q . Then there are matrices A ∈ M(n × n; E q ) and B ∈ M(n × m; E q ) such that (( A|B)id)(x, u)=g(x, u) ∀ (x, u) ∈ F n q ×F m q where (A|B) ∈ M(n ×(n + m); E q ) is the matrix that results by writing A and B side by side. In this sense we denote g as the monomial control system (A, B) with n state variables and m control inputs. Proof. This follows immediately from the previous definitions. Remark 61. If the matrix B ∈ M (n ×m; E q ) is equal to the zero matrix, then g is called a control system with no controls. In contrast to linear control systems (see the previous sections and also Sontag (1998)), when the input vector u ∈ F m q satisfies u =  1:=(1, , 1) t ∈ F m q then no control input is being applied on the system, i.e. the monomial dynamical system over F q σ : F n q → F n q x → g(x,  1) satisfies σ (x)=((A|0)id)(x, u) ∀ (x, u) ∈ F n q ×F m q where 0 ∈ M(n × m; E q ) stands for the zero matrix. Definition 62. Let F q be a finite field and n, m ∈ N natural numbers. A monomial feedback controller is a mapping f : F n q → F m q such that for every i ∈{1, , m} there is a tuple (F i1 , , F in ) ∈ E n q such that f i (x)=x F i1 1 x F in n ∀ x ∈ F n q Remark 63. We exclude in the definition of monomial feedback controller the possibility that one of the functions f i is equal to the zero function. The reason for this will become apparent in the next remark (see below). Now we are able to formulate the first control theoretic problem to be addressed in this section: Problem 64. Let F q be a finite field and n, m ∈ N natural numbers. Given a monomial control system g : F n q × F m q → F n q with completely observable state, design a monomial state feedback controller f : F n q → F m q such that the closed-loop system h : F n q → F n q x → g( x, f (x)) 471 Discrete Time Systems with Event-Based Dynamics: Recent Developments in Analysis and Synthesis Methods has a desired period number and cycle structure of its phase space. What properties has g to fulfill for this task to be accomplished? Remark 65. Note that every component h i : F n q → F q , i = 1, , n x → g i (x, f (x)) is a nonzero monic monomial function, i.e. the mapping h : F n q → F n q is a monomial dynamical system over F q . Remember that we excluded in the definition of monomial feedback controller the possibility that one of the functions f i is equal to the zero function. Indeed, the only effect of a component f i ≡ 0 in the closed-loop system h would be to possibly generate a component h j ≡ 0. As explained in Remark 28 of Section 3.1, this component would not play a crucial role determining the long term dynamics of h. Due to the monomial structure of h, the results presented in Section 3.1 of this chapter can be used to analyze the dynamical properties of h. Moreover, the following identity holds h =(A + B · F)id where F ∈ M(m ×n; E q ) is the corresponding matrix of f (see Remark 30), (A, B) are the matrices in Lemma 60 and id ∈ F n n (F q ). To see this, consider the mapping μ : F m q → F n q u → g(  1, u) where  1 ∈ F n q . From the definition of g it follows that μ ∈ MF n m (F q ). Now, since f ∈ MF m n (F q ), by Remark 30 we have for the composition μ ◦ f : F n q → F n q μ ◦ f =(B · F)id Now its easy to see h =(A + B · F)id The most significant results proved in Colón-Reyes et al. (2004), Delgado-Eckert (2008) concern Boolean monomial dynamical systems with a strongly connected dependency graph. Therefore, in the next section we will focus on the solution of Problem 64 for Boolean monomial control systems g : F n 2 ×F m 2 → F n 2 with the property that the mapping σ : F n 2 → F n 2 x → g( x,  1) has a strongly connected dependency graph. Such systems are called strongly dependent monomial control systems. If we drop this requirement, we would not be able to use Theorems 45 and 46 to analyze h regarding its cycle structure. However, if we are only interested in forcing the period number of h to be equal to 1, we can still use Theorem 47 (see Remark 48). This feature will be exploited in Section 3.3, when we study the stabilization problem. Although the above representation h =(A + B · F)id 472 Discrete Time Systems of the closed loop system displays a striking structural similarity with linear control systems and linear feedback laws, our approach will completely differ from the well known "Pole-Assignment" method. 3.3 State feedback controller design for Boolean monomial control systems Our goal in this section is to illustrate how the loop number, a parameter that, as we saw, characterizes the dynamic properties of Boolean monomial dynamical systems, can be exploited for the synthesis of suitable feedback controllers. To this end, we will demonstrate the basic ideas using a very simple subclass of systems that allow for a graphical elucidation of the rationale behind our approach. The structural similarity demonstrated in Remark 53 then enables the extension of the results to more general cases. A rigorous implementation of the ideas developed here can be found in Delgado-Eckert (2009b). As explained in Remark 53, a Boolean monomial dynamical system with a strongly connected non-trivial dependency graph can be visualized as a simple cycle of loop-equivalence classes (see Fig. 1). In the simplest case, each loop-equivalence class only contains one node and the dependency graph is a closed path. A first step towards solving Problem 64 for strongly dependent Boolean monomial control systems g : F n 2 × F m 2 → F n 2 would be to consider the simpler subclass of problems in which the mapping σ : F n 2 → F n 2 x → g(x,  1) simply has a closed path of length n as its dependency graph (see Fig. 2 a for an example in the case n = 6). By the definition of dependency graph and after choosing any monomial feedback controller f : F n 2 → F m 2 , it becomes apparent that the dependency graph of the closed-loop system h f : F n 2 → F n 2 x → g(x, f (x)) arises from adding new edges to the dependency graph of σ. Since we assumed that the dependency graph of σ is just a closed path, adding new edges to it can only generate new closed paths of length in the range 1, . . . , n −1. By Corollary 41, we immediately see that the loop number of the modified dependency graph (i.e., the dependency graph of h f ) must be a divisor of the original loop number. This result is telling us that no matter how complicated we choose a monomial feedback controller f : F n 2 → F m 2 , the closed loop system h f will have a dependency graph with a loop number L  which divides the loop number L of the dependency graph of σ. This is all we can achieve in terms of loop number assignment. When a system allows for assignment to all values out of the set D (L), we call it completely loop number controllable. We just proved this limitation for systems in which σ has a simple closed path as its dependency graph. However, due to the structural similarity between such systems and strongly dependent systems (see Remark 53), this result remains valid in the general case where σ has a strongly connected dependency graph. Let us simplify the scenario a bit more and assume that the system g has only one control variable u (i.e., g : F n 2 × F 2 → F n 2 ) and that this variable appears in only one component function, say g k . As before, assume σ has a simple closed path as its dependency graph. Under these circumstances, we choose the following monomial feedback controllers: f i : F n 2 → F 2 , 473 Discrete Time Systems with Event-Based Dynamics: Recent Developments in Analysis and Synthesis Methods f i (x) := x i , i = 1, , n. When we look at the closed-loop systems h f i : F n 2 → F n 2 x → g( x, f i (x)) and their dependency graphs, we realize that the dependency graph of h f i corresponds to the one of σ with one single additional edge. Depending on the value of i under consideration, this additional edge adds a closed path of length l in the range l = 1, , n −1 to the dependency graph of σ. In Figures 2 b-e, we see all the possibilities in the case of n = L = 6, except for l = 1 (self-loop around the kth node). L = 6 L = 2 L = 3 L = 1 L = 1 L = 1 a b c d e f Fig. 2. Loop number assignment through the choice of different feeback controllers. We realize that with only one control variable appearing in only one of the components of the system g, we can set the loop number of the closed-loop system h f i to be equal to any of the possible values (out of the set D (L)) by choosing among the feedback controllers f i , i = 1, , n, defined above. This proves that the type of systems we are considering here are indeed completely loop number controllable. Moreover, as illustrated in Figure 2 f, if the control variable u would appear in another component function of g, we may loose the loop number controllability. Again, due to the structural similarity (see Remark 53), this complete loop number controllability statement is valid for strongly dependent systems. In the light of Theorem 47 (see Remark 48), for the stabilization 13 problem we can consider arbitrary Boolean monomial control systems g : F n 2 × F m 2 → F n 2 , maybe only requiring the obvious condition that the mapping σ is not already a fixed point system. Moreover, the statement of Theorem 47 is telling us that such a system will be stabilizable if and only if the component functions g j depend in such a way on control variables u i , that every strongly connected component of the dependency graph of σ can be forced into loop number one by incorporating suitable additional edges. This corresponds to the choice of a suitable feedback controller. The details and proof of this stabilizability statement as well as a brief description of a stabilization procedure can be found in Delgado-Eckert (2009b). 13 Note that in contrast to the definition of stability introduced in Subsection 1.2.1, in this context we refer to stabilizability as the property of a control system to become a fixed point system through the choice of a suitable feedback controller. 474 Discrete Time Systems 4. Conclusions In this chapter we considered discrete event systems within the paradigm of algebraic state space models. As we pointed out, traditional approaches to system analysis and controller synthesis that were developed for continuous and discrete time dynamical systems may not be suitable for the same or similar tasks in the case of discrete event systems. Thus, one of the main challenges in the field of discrete event systems is the development of appropriate mathematical techniques. Finding new mathematical indicators that characterize the dynamic properties of a discrete event system represents a promising approach to the development of new analysis and controller synthesis methods. We have demonstrated how mathematical objects or magnitudes such as invariant polynomials, elementary divisor polynomials, and the loop number can play the role of the aforementioned indicators, characterizing the dynamic properties of certain classes of discrete event systems. Moreover, we have shown how these objects or magnitudes can be used to effectively address controller synthesis problems for linear modular systems over the finite field F 2 and for Boolean monomial systems. We anticipate that the future development of the discrete event systems field will not only comprise the derivation of new mathematical methods, but also will be concerned with the development of efficient algorithms and their implementation. 5. 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On self-stabilizing systems: an approach to the specification and design of fault tolerant systems, Decision and Control, IEEE Conference on, pp. 1200 –1205 vol.2. 476 Discrete Time Systems Mihaela Neam¸tu and Dumitru Opri¸s W est University of Timi¸soara Romania 1. Introduction The dynamical systems with discrete time and delay are obtained by the discretization of the systems of differential equations with delay, or by modeling some processes in which the time variable is n ∈ IN and the state variables at the moment n −m,wherem ∈ IN , m ≥ 1, are taken into consideration. The processes from this chapter have as mathematical model a system of equations given by: x n+1 = f (x n , x n−m , α),(1) where x n = x(n) ∈ IR p , x n−m = x(n − m) ∈ IR p , α ∈ IR and f : IR p × IR p × IR → IR p is a seamless function, n, m ∈ IN with m ≥ 1. The properties of function f ensure that there is solution for system (1). The system of equations (1) is called system with discrete-time and delay. The analysis of the processes described by system (1) follows these steps. Step 1. Modeling the process. Step 2. Determining the fixed points for (1). Step 3. Analyzing a fixed point of (1) by studying the sign of the characteristic equation of the linearized equation in the neighborhood of the fixed point. Step 4. Determining the value α = α 0 for which the characteristic equation has the roots μ 1 (α 0 )=μ(α 0 ), μ 2 (α 0 )= μ(α 0 ) with their absolute value equal to 1, and the other roots with their absolute value less than 1 and the following formulas: d |μ(α)| dα    α=α 0 = 0, μ(α 0 ) k = 1, k = 1, 2, 3, 4 hold. Step 5. Determining the local center manifold W c loc (0): y = zq + z q + 1 2 w 20 z 2 + w 11 zz + 1 2 w 02 z 2 + where z = x 1 + ix 2 ,with(x 1 , x 2 ) ∈ V 1 ⊂ IR 2 ,0∈ V 1 , q an eigenvector corresponding to the eigenvalue μ (0) and w 20 , w 11 , w 02 are vectors that can be determined by the invariance Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications 26 condition of the manifold W c loc (0) with respect to the transformation x n−m = x 1 , , x n = x m , x n+1 = x m+1 . The restriction of system (1) to the manifold W c loc (0) is: z n+1 = μ(α 0 )z n + 1 2 g 20 z 2 n + g 11 z n z n + 1 2 g 02 z 2 n + g 21 z 2 n z n /2, (2) where g 20 , g 11 , g 02 , g 21 are the coefficients obtained using the expansion in Taylor series including third-order terms of function f . System (2) is topologically equivalent with the prototype of the 2-dimensional discrete dynamic system that characterizes the systems with a Neimark–Sacker bifurcation. Step 6. Representing the orbits for system (1). The orbits of system (1) in the neighborhood of the fixed point x ∗ are given by: x n = x ∗ + z n q + ¯ z n ¯ q + 1 2 r 20 z 2 n + r 11 z n ¯ z n + 1 2 r 02 ¯ z 2 n (3) where z n is a solution of (2) and r 20 , r 11 , r 02 are determined with the help of w 20 , w 11 , w 02 . The properties of orbit (3) are established using the Lyapunov coefficient l 1 (0).Ifl 1 (0) < 0 then orbit (3) is a stable invariant closed curve (supercritical) and if l 1 (0) > 0thenorbit(3)is an unstable invariant closed curve (subcritical). The perturbed stochastic system corresponding to (1) is given by: x n+1 = f (x n , x n−m , α)+g(x n , x n−m )ξ n ,(4) where x n = x 0 n , n ∈ I = {−m, −m + 1, , −1, 0} is the initial segment to be F 0 -measurable, and ξ n is a random variable with E(ξ n )=0, E(ξ 2 n )=σ > 0andα is a real parameter. System (4) is called discrete-time stochastic system with delay. For the stochastic discrete-time system with delay, the stability in mean and the stability in square mean for the stationary state are done. This chapter is organized as follows. In Section 2 the discrete-time deterministic and stochastic dynamical systems are defined. In Section 3 the Neimark-Sacker bifurcation for the deterministic and stochastic Internet control congestion with discrete-time and delay is studied. Section 4 presents the deterministic and stochastic economic games with discrete-time and delay. In Section 5, the deterministic and stochastic Kaldor model with discrete-time is analyzed. Finally some conclusions and future prospects are provided. For the models from the above sections we establish the existence of the Neimark-Sacker bifurcation and the normal form. Then, the invariant curve is studied. We also associate the perturbed stochastic system and we analyze the stability in square mean of the solutions of the linearized system in the fixed point of the analyzed system. 2. Discrete-time dynamical systems 2.1 The definition of the discrete-time, deterministic and stochastic systems We intuitively describe the dynamical system concept. We suppose that a physical or biologic or economic system etc., can have different states represented by the elements of a set S. These states depend on the parameter t called time. If the system is in the state s 1 ∈ S, at the moment t 1 and passes to the moment t 2 in the state s 2 ∈ S, then we denote this transformation by: Φ t 1 ,t 2 (s 1 )=s 2 478 Discrete Time Systems [...]... ¯ P2 (λ) = det(λI − A) For system (8), respectively (9), the analysis of the solutions can be done by studying the roots of the equation P2 (λ) = 0 482 Discrete Time Systems 2.3 Discrete- time dynamical systems with one parameter Consider a discrete- time dynamical system depending on a real parameter α, defined by the application: R R (11) x → f ( x, α), x ∈ I m , α ∈ I where f : I m × I → I m is a seamless... g11 , g02 , g21 are the coefficients obtained using the expansion in Taylor series including third-order terms of function f 484 Discrete Time Systems 2.4 The Neimark-Sacker bifurcation for a class of discrete- time dynamical systems with delay A two dimensional discrete- time dynamical system with delay is defined by the equations x n +1 = x n + f 1 ( x n , y n , α ) (13) y n +1 = y n + f 2 ( x n − m... Elements used for the study of the discrete- time dynamical systems Consider the following discrete- time dynamical system defined on I m : R x n +1 = f ( x n ), n∈I N (6) R where f : I m → I m is a Cr class function, called vector field R Some information, regarding the behavior of (6) in the neighborhood of the fixed point, is obtained studying the associated linear discrete- time dynamical system R Let x0... dynamical system with discrete time on I m , is the homomorphism of R groups Φ : ( Z, +) → ( Di f f r ( I m ), ◦) R The orbit through x0 ∈ I m of a dynamical system with discrete- time is: O f ( x0 ) = { , f −( n) ( x0 ), , f (−1) ( x0 ), x0 , f ( x0 ), , f ( n) ( x0 ), } = { f ( n) ( x0 )}n∈ Z Thus O f ( x0 ) represents a sequences of images of the studied process at regular periods of time For the study... with discrete time, the structure of the orbits’set is R analyzed For a dynamical system with discrete time with the initial condition x0 ∈ I m (m = 1, 2, 3) we can represent graphically the points of the form xn = f n ( x0 ) for n iterations of the thousandth or millionth order Thus, a visual geometrical image of the orbits’set structure is created, which suggests some properties regarding the particularities... be approved or disproved by theoretical or practical arguments An explicit dynamical system with discrete time has the form: x n +1 = f ( x n − p , x n ), n∈I N, (5) where f : I m × I m → I m , xn ∈ I m , p ∈ I is fixed, and the initial conditions are x− p , x1− p , R R R R N R , x0 ∈ I m 480 Discrete Time Systems For system (5), we use the change of variables x1 = xn− p , x2 = xn−( p−1) , , x p = xn−1... j = 1 m is called the linear discrete- time dynamical system associated to (6) and the fixed point x0 = f ( x0 ) If the characteristic polynomial of D f ( x0 ) does not have roots with their absolute values equal to 1, then x0 is called a hyperbolic fixed point We have the following classification of the hyperbolic fixed points: Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications... m+1 ∂x m+1 ∂x m+1 (16) Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications b20 = b02 = b21 ∂2 g (0, 0, α), ∂x1 ∂x1 ∂2 g ∂x m+2 ∂x m+2 b11 = (0, 0, α), b30 = ∂3 g = 1 1 m+2 (0, 0, α), ∂x ∂x ∂x b03 = ∂3 g ∂x m+2 ∂x m+2 ∂x m+2 b12 = ∂2 g ∂x1 ∂x m+2 (0, 0, α), ∂3 g ∂x1 ∂x1 ∂x1 ∂3 g 485 (0, 0, α), ∂x1 ∂x m+2 ∂x m+2 (17) (0, 0, α), (0, 0, α) With (16) and (17) from (15) we have:... with the notations from (35) (iii) Because μ2 , μ2 , 1 are not roots of the characteristic equation (19) then the linear systems (34) are determined compatible systems The relations (37) are obtained by simple calculation (iv) From (33) with (35) we obtain (38) Consider the discrete- time dynamical system with delay given by (13), for which the roots of the characteristic equation satisfy the hypotheses... discretizing system (42) is given by: xn+1 = xn − ak f ( xn−m ) + kw (43) N, for n, m ∈ I m > 0 and it represents the dynamical system with discrete- time for Internet congestion with one link and a single source Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications 491 Using the change of variables x1 = xn−m , , x m = xn−1 , x m+1 = xn , the application associated to . and discrete time dynamical systems may not be suitable for the same or similar tasks in the case of discrete event systems. Thus, one of the main challenges in the field of discrete event systems. equation P 2 (λ)=0. 481 Discrete Deterministic and Stochastic Dynamical Systems with Delay - Applications 2.3 Discrete- time dynamical systems with one parameter Consider a discrete- time dynamical system. in the fixed point of the analyzed system. 2. Discrete- time dynamical systems 2.1 The definition of the discrete- time, deterministic and stochastic systems We intuitively describe the dynamical